Knots and 4-manifolds

TL;DR

This paper discusses the properties of slice knots, methods for analyzing their diagrams, and presents a computer search for slice alternating knots, contributing to understanding knot sliceness in 4-manifolds.

Contribution

It introduces new techniques for diagramming double branched covers and surfaces, and reports on a computer search for slice alternating knots.

Findings

Methods for drawing diagrams of double branched covers

Criteria for determining if a knot is slice

Results from the computer search for slice alternating knots

Abstract

These notes are based on the lectures given by the author during Winter Braids IX in Reims in March 2019. We discuss slice knots and why they are interesting, as well as some ways to decide if a given knot is or is not slice. We describe various methods for drawing diagrams of double branched covers of knots in the 3-sphere and surfaces in the 4-ball, and how these can be useful to decide if an alternating knot is slice. We include a description of the computer search for slice alternating knots due to the author and Frank Swenton.